We may assume \(\Vert \Omega \Vert _{L^{\infty }(\mathbb {S}^{d-1})}=1\)  and denote by \(t\)  the dual exponent of \(\beta \) .
It is enough to show for each \(r\ge 1\)  and \(t=2r\) ,
\(\frac{1}{|Q|^{\frac{1}{t}}} \Big \Vert  \sum _{\nu } \sum _sG_{\nu }^j(I-P_{s-j\kappa })T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} \Big \Vert _{L^{t}(\mathbb {R}^d)}\lesssim 2^{-j\gamma /2} \Vert b\Vert _{\dot{\mathcal {X}}_1(\mathcal {Q})}.\) 
 Denote \(M_\nu = \sum \limits _s {G_{\nu }^j}(I-P_{s-j\kappa } )T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} \) . Applying Plancherel's theorem, we get
\( \Big \Vert \sum \limits _{\nu _1,\cdots ,\nu _r} \prod \limits _{k=1}^{r} M_{\nu _k}\Big \Vert ^2_{L^{2}(\mathbb {R}^d)}&\lesssim 2^{j\gamma (d-2)r}\sum \limits _{\nu _1,\cdots ,\nu _r} \Big \Vert  \prod \limits _{k=1}^{r} \sum \limits _{s}(I-P_{s-j\kappa } )T^{j}_{\Omega ,{L};\,s,\nu }b_{s-j} \Big \Vert _{L^{2}(\mathbb {R}^d)} ^2\\&\lesssim 2^{j\gamma (d-2)r} 2^{j\gamma (d-1)r}\Big \Vert  \prod \limits _{k=1}^{r} \sum \limits _{s}(I-P_{s-j\kappa } )T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} \Big \Vert _{L^{2}(\mathbb {R}^d)} ^2\nonumber \\& \lesssim 2^{-j\gamma r}2^{j\gamma (d-1)t} \sup \limits _{\nu } \Big \Vert  \sum \limits _{s}(I-P_{s-j\kappa } )T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} \Big \Vert _{L^{t}(\mathbb {R}^d)} ^t.\nonumber \) 
 On the other hand, inequality (REF ) tells us that
\(\big | (I-P_{s-j\kappa } )T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} (x)\big |\lesssim jH_{s,\nu }^j*|b_{s-j}|(x),\) 

where \( H_{s,\nu }^j(x)=2^{-sd}\chi _{\mathcal {R}_{s\nu }^j}(x)\)  is defined in Section 3.
Following the proof of (4.6) in [1]}, for each fixed \(\nu \) , one may easily verify that
\(\sum \limits _{s_1\ge \cdots \ge s_t}\int \bigg (\prod _{k=1}^{t}H_{s_k,\nu }^j (x-y_k) |b_{s_k-j}(y_k) |\bigg ) dy_1\cdots dy_t dx\lesssim 2^{-j\gamma (d-1)t} |Q| \Vert b\Vert ^t_{\dot{\mathcal {X}}_1(\mathcal {Q})}. \) 
Combining the above estimate (REF ) with (REF ), one may obtain
\(& \Big \Vert  \sum _{\nu } \sum _sG_{\nu }^j(I-P_{s-j\kappa })T^{j}_{\Omega ,{\mathbb {L}};\,s,\nu }b_{s-j} \Big \Vert ^t_{L^{t}(\mathbb {R}^d)} \lesssim 2^{-j\gamma r}j |Q| \Vert b\Vert ^t_{\dot{\mathcal {X}}_1(\mathcal {Q})}.\) 
This finishes the proof of Lemma REF .
